The matrix-valued numerical range over finite fields

Abstract

In this paper we define and study the matrix-valued $k\times k$ numerical range of $n\times n$ matrices using the Hermitian product and the product with $n\times k$ unitary matrices $U$ (on the right with $U$, on the left with its adjoint $U^\dagger = U^{-1}$). For all $i, j=1,\dots ,k$ we study the possible $(i,j)$-entries of these $k\times k$ matrices. Our results are for the case in which the base field is finite, but the same definition works over $\mathbb {C}$. Instead of the degree $2$ extension $\mathbb {R}\hookrightarrow \mathbb {C}$ we use the degree $2$ extension $\mathbb {F} _q\hookrightarrow \mathbb {F} _{q^2}$, $q$ a prime power, with the Frobenius map $t\mapsto t^q$ as the nonzero element of its Galois group. The diagonal entries of the matrix numerical ranges are the scalar numerical ranges, while often the nondiagonal entries are the entire $\mathbb {F} _{q^2}$. We also define the matrix-valued numerical range map.